Uncertainty Quantification has been studied in the last decades on random partial differential equations involving Gaussian random fields. Gaussian random fields are easy to simulate but they offer only little distributional flexibility and are by nature continuous. In this talk we define subordinated Random Fields and apply them to an elliptic equation. Subordinated fields (inspired by the corresponding subordinated processes) can have discontinuities in their paths and depending on the choice of subordinator and the subordinated field various pointwise distributions can be realized. Next to providing theoretical results on distributional properties of the fields we present different methods to numerically calculate moments of the resulting random PDE's.